Yifeng Xu

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44 Questions

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19	Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?
13	Showing that $ \|\cos x\|+\|\cos 2x\|+\cdots+\|\cos 2^nx\|\geq \dfrac{n}{2\sqrt{2}}$
9	Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$.
9	Prove that $\int_0^1\|f''(x)\|dx\ge4.$
7	Geometric inequality with a triangle

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900 Reputation

-2	Prove that $\int_0^1[f''(x)]^2dx\ge4.$
+5	Showing that $ \|\cos x\|+\|\cos 2x\|+\cdots+\|\cos 2^nx\|\geq \dfrac{n}{2\sqrt{2}}$
+15	Show that $\lim_{n\to\infty}n\int_0^1f(x)g(x^n)dx=f(1)\int_0^1\frac{g(x)}{x}dx$
+5	Evaluate $\int_C\frac{dw}{e^w-1}$ over some loop C contained in the annulus $0<\|z\|<2\pi$.

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3 Answers

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0	Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$.
0	$\sum_{i=1}^nx_i^2+2\sum_{1\le k <j\le n}\sqrt{\frac{k}{j}}x_kx_j=1$, find the maximum and minimum of $\sum_{i=1}^n x_i.$
-1	Prove that for all positive reals $k,j$ with $2k>3j$, there exists positive integers $s,t$ such that $k > \|\sqrt{s}-\sqrt{t}\| > j$

45 Tags

0 contest-math × 28	0 optimization × 4
0 analysis × 24	0 sequences-and-series × 4
0 calculus × 18	0 linear-algebra × 3
0 limit × 7	0 definite-integral × 3
0 integral × 5	0 elementary-number-theory × 2

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$Mathematics Stack Exchange$	Mathematics	900 rep	9
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